The document discusses first-order partial differential equations (PDEs), defining them as equations involving a function of multiple independent variables and their partial derivatives. It explains the concepts of order and degree of a PDE, as well as the conditions for forming a PDE through the elimination of arbitrary constants or functions. General rules are provided regarding the relationship between the number of arbitrary constants and the independent variables in determining the order of the resulting PDE.

1. By
Dr. B.M.B.Krushna
Sr. Asst. Professor
Dept. Of Mathematics
MVGR COLLEGE OF ENGINEERING (A)
1
FIRST ORDER PARTIAL DIFFERENTIAL
EQUATIONS

2.  A partial differential equation(shortly., PDE) is
an equation for a function which depends on
more than one independent variable which
involves the independent variables, the
function, and partial derivatives of the
function.

3. 

4.  The order of a PDE is defined as the order of
the highest order derivative occur in the
equation.
 The degree of a PDE is defined as the +ve
integral power to which the highest order
derivative occur in the equation.

5. 

6. We can form a PDE in Two ways
 Elimination of arbitrary constants from a
relation f(x,y,z,a,b,…)=0
 Elimination of arbitrary functions

7.  General rules to remember.
 If the number of arbitrary constants to be
eliminated is equal to the number of
independent variables, then we get a PDE
of first order.
 If the number of arbitrary constants to be
eliminated is greater than the number of
independent variables, then we get a PDE
of 2nd or higher order. In this case we may
get more than one PDE.

8.  If the number of arbitrary constants to be
eliminated is less than the number of
independent variables, then we get a PDE of
first order.
In this case we may get more than one PDE.
Note. In this chapter we use z=f(x,y), where x,y
are independent variables and z is dependent
variable.

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